Types of quantum field thories
A Green-Schwarz mechanism (named after Michael Green and John Schwarz) is a modification of an action functional of a quantum field theory involving higher gauge fields that makes a quantum anomaly of the original action functional disappear.
More in detail:
An action functional in path integral quantization is said to be anomalous if it is only locally identified with a function on the configuration space of fields, but is globally instead a section of a line bundle (usually equipped with connection).
Given two anomalous action functionals in this sense, it may happen that while the two corresponding line bundles on configuration space are each nontrivial, their tensor product becomes trivializable. In this case one can consider the non-anomalous action function given by the sum of the two anomalous action functionals. This is what is called anomaly cancellation of one piece of an action functional against another.
The two main sources of examples for action functionals that are anomalous are
the standard fermionic action functional? (see there for details) for chiral fermions
A Green–Schwarz mechanism is the addition of an action functional for higher differential cocycles with magnetic charges such that their quantum anomaly cancels a given Pfaffian line bundle: so it is a choice of by itself ill-defined action functional for higher gauge theory that cancels the ill-definedness of an action functional for chiral fermions.
In the more strict and original sense of the word, the Green–Schwarz mechanism is the application of this procedure in the theory called heterotic supergravity: there it so happens that the Pfaffian line bundle of the fermionic action has as Chern class the transgression of a degree-12 class in ordinary differential cohomology that factorizes as . Since heterotic supergravity contains a higher gauge field that couples to strings, this is precisely of the form that the anomaly for the corresponding higher gauge theory in the presence of magnetic charges gives rise to. So the original Green–Schwarz anomaly cancellation mechanism consist of modifying the “naive” action functional for heterotic supergravity by adding the contribution that corresponds to adding a magnetic current of the form
in degree this is an electromagnetic field;
in degree this is a Kalb-Ramond field;
in degree a supergravity C-field.
that sends the field
to the exponential of the integral over the spacetime of the differential form obtained as the wedge product of the curvature form with its image under the Hodge star operator correspondonding to the pseudo-Riemannian metric on .
The fact that this map function defined in terms of cocycles is a well defined function on cohomology means that the action functional is gauge invariant. At this point this is just the trivial statement that under a gauge transformation
the curvature invariant: .
The curvature of is the electric current form. The action functional that encodes the force of the gauge field exerted on this electric charge distribution is locally on coordinate charts given by the integral , where is the local connection -form of the gauge fiedl .
Globally, this contribution is given by the push-forward
We now consider one more additional term in the action functional, one that desribes moreover the interaction of our gauge field with a distribution of -magnetic charge on , in addition to the interaction with the distribution of electric charge described above.
The magnetic charge distribution itself is also modeled as a cocycle in ordinary differential cohomology. As opposed to the electric charge it is however not part of the dynamics but of the kinemtics of the system: it does not manifestly show up in the integral expression for the action functional, but does modify the nature of the configuration space that this action functional is defined on.
Namely the kinematic higher Maxwell equations is a condition of the form
where is the magnetic charge distribution. If is the curvature of a circle n-bundle with connection, then necessarily . Therefore the system of higher electromagnetism in the presence of magnetic charge cannot be modeled any more by cocycles in ordinary differential cohomology.
We shall write for the ∞-groupoid of twisted cocycles for this fixed twist. The crucial point is now the following:
the above expression
for the electric coupling can still be given sense, even with , but it no longer has the interpretation of a circle group-valued function. Rather, it has now the interpretation of a section of a line bundle
In the next section we formalize properly the notion of this line bundle on configuration space.
In order to formalize this we have to refine the formalization of the structure of the configuration space. So far we had regarded the set of gauge equivalence classes of field configurations. This is the set of connected components of the full cocycle ∞-groupoid
objects are field configurations on ;
2-morphisms are gauge transformations of gauge transformation,
and so on.
for this smooth -groupoid of configuration of the physical system – defined as the internal hom in terms of the closed monoidal structure on the (∞,1)-topos Smooth∞Grpd of into the target object of the higher gauge theory, (this object is discussed in detail here; it is presented under the Dold-Kan correspondence by the Deligne complex of sheaves on CartSp).
of gauge fields on the product of spacetime with the parameter space . (See for instance Lie integration and connection on an ∞-bundle for details on how differential forms on encode -families of forms on ).
This way the configuration space of higher electromagnetism in the presence of electric and magnetic charge is naturally incarnated as an object in the cohesive (∞,1)-topos of smooth ∞-groupoids, and accordingly all the differential geometric structures in cohesive -topos are available. In particular we may speak of line bundle with connection on , gevin for instance by morphisms
the underlying class in ordinary cohomology
is the anomaly of the system of higher electromagnetism couple to electic and magnetic charge;
its curvature 2-form
is the differential anomaly.
This means that for every parameter space SmoothMfd and every morphism – which corresponds by the nature of the ∞-stack to a field configuration – the pullback of this differential form to yields the ordinary differential form in the image of under the fiber integration map
We can now state the Green-Schwarz mechanism itself.
and equipped with an action functional
that is a section of an anomaly line bundle
such that the curvature 2-form of happens to be of the form
for some and .
Then the Green-Schwarz mechanism is the map that changes this physical system by adding magnetic charge to it, given by a cocycle with
This means by the above that the new action functional is now a section
of the tensor product of the two anomaly line bundles. The Chern class of the tensor product is the sum of the two Chern-classes, hence by definition of they cancel, so that is trivializatable as a line bundle with connection.
A choice of such trivialization identifies the section then with an ordinary function
This is the anomaly-free action functional after the Green-Schwarz mechanism has been applied.
The configurations of this theory are given by
The path integral over the fermionic part of the action
where the ellipses indicate decomposable curvature characteristic forms.
Therefore in this case the Green-Schwarz mechanism consists of
adding to the system fivebrane magnetic charge given by .
adding to the system string electric charge .
This means that to the action functional is added the factor
which is locally on given in the exponent by the integral
The nature of the field configuration obtained this way – spin connection with twist of th Kalb-Ramond field by the Pontryagin class – may be understood conciesely as constituting a twisted differential string structure on . See there for more details.
A clear and precise account of what anomalies are and what the Green-Schwarz mechanism is to cancel them is given in
For the moment, see there for further references.
An account of historical developments is in section 7 of
The full formula for the differential form data including the fermionic contributions is in
and references given there.